An Asymptotic Expansion for the Distribution of Euclidean Distance-Based Discriminant Function in Normal Populations

This paper presents an asymptotic expansion for the distribution of the Euclidean distance-based discriminant function under two normal populations. It is well known that the Euclidean distance-based classifier has good discriminant performance for high-dimensional data. On the other hand, there are few evaluations of the performance of its discrimination in low dimensions. Thus, deriving the asymptotic expansion for the Euclidean distance-based classifier is of importance in evaluating the performance to compare with other classifiers. For applications, we compare the two discriminant functions by using their asymptotic expansions under normal populations. We show a case that the Euclidean distance-based discriminant function is better than a Fisher's linear discriminant function regardless of the sample sizes under a certain mean and covariance structure.